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Stability estimates for reconstruction from the Fourier transform on the ball

Mikhail Isaev, Roman Novikov

To cite this version:

Mikhail Isaev, Roman Novikov. Stability estimates for reconstruction from the Fourier transform on the ball. Journal of Inverse and Ill-posed Problems, De Gruyter, 2021, 29 (3), pp.421-433. �hal- 02906953�

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Stability estimates for reconstruction from the Fourier transform on the ball

Mikhail Isaev

School of Mathematics Monash University Clayton, VIC, Australia mikhail.isaev@monash.edu

Roman G. Novikov

CMAP, CNRS, Ecole Polytechnique Institut Polytechnique de Paris

Palaiseau, France IEPT RAS, Moscow, Russia novikov@cmap.polytechnique.fr

Abstract

We prove H¨older-logarithmic stability estimates for the problem of finding an integrable function v on Rd with a super-exponential decay at infinity from its Fourier transform Fv given on the ball Br. These estimates arise from a H¨older- stable extrapolation of Fv from Br to a larger ball. We also present instability examples showing an optimality of our results.

Keywords: ill-posed inverse problems, H¨older-logarithmic stability, exponential instability, Chebyshev extrapolation

AMS subject classification: 42A38, 35R30, 49K40

1 Introduction

We consider the classical Fourier transform F defined by Fv(ξ) := 1

(2π)d

Z

Rd

eiξxv(x)dx, ξRd,

The first author’s research is supported by the Australian Research Council Discovery Early Career Researcher Award DE200101045.

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where v is a test function onRd and d1. Let Br :=

ξ Rd:|ξ|< r , wherer > 0.

Assume that v is integrable and, for some N, σ >0 and ν 1, we have that Qv(λ) := 1

(2π)d

Z

Rd

eλ|x||v(x)|dxNexp (σλν), for all λ0. (1.1)

Remark 1.1. In particular, for the case of ν = 1, the class of functions satisfying (1.1) includes all functions v with suppv Bσ and (2π)1 dkvkL1(Rd) N. Furthermore, if v is such that

|v(x)| ≤Cexp (−µ|x|η) for some µ, C >0 and η >1,

then assumption (1.1) holds with ν := η−1η and with some positive constants σ=σ(µ, η) and N =N(C, µ, η, d).

Under assumption (1.1), we consider the following two problems:

Problem 1.1. GivenFv on the ball Br. Findv.

Problem 1.2. GivenFv on the ball Br. FindFv onBR, where R > r.

Problems 1.1 and 1.2 are fundamental in the theory of inverse coefficient problems. For example, Problem 1.1 with r = 2

E can be regarded as a linearized inverse scattering problem for the Schr¨odinger equation with potential v at fixed positive energy E, for d 2, and on the the energy interval [0, E], for d 1. More details can be found in [12, Section 4]. Problem 1.1 with r = ω0 also arises in a multi-frequency inverse source problem for the homogeneous Helmholtz equation with frequenciesω [0, ω0]; see Bao et al. [2, Section 3] for more details. In addition, in many cases, Problem 1.2 is an essential step for solving Problem 1.1. For more applications related to Problems 1.1 and 1.2 in the case of compactly supported v, see [8] and references therein.

The present work continues the studies of our recent article [8], which considers the case of compactly supported functions v. Besides, in [8], we deal with reconstructions of Fv on [−R, R]d and v onRd fromFv given on the cube [−r, r]d, in place of the balls BR andBr. Due to the equivalence ofk·k2-norm andk·k-norm inRd, these formulations are essentially equivalent, butBRandBrare more natural in the context of inverse problems.

In the present work, under assumption (1.1), we give H¨older-logarithmic stability estimates for Problem 1.1 in the norm of L(Rd) and of Hs(Rd), for any real s; see Section 3. (Note that the stability estimates of [8] are given in the norm of L2(Rd)

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only.) In addition, we obtain H¨older stability estimates for Problem 1.2 in the norm of L(BR); see Section 5. The related reconstruction procedures are also given; see Sections 2 and 3. Besides, we present examples showing an optimality of our stability estimates and reconstruction procedures; see Section 4.

2 Reconstruction procedures

LetF−1 be the classical inverse Fourier transform defined by F−1[u](x) :=

Z

Rd

u(ξ)e−iξxdξ, xRd.

For a given r > 0, we consider the following family of extrapolations CR,n : L(Br) L(BR), depending on two parameters R r and n N:={0,1, . . .}. For a function w onBr (for example, such that w≈ Fv|Br), we define

[CR,nw](ξ) :=

w(ξ), ξBr,

n−1

X

k=0

ak

ξ

|ξ|

Tk

|ξ|

r

, ξBR\Br,

0, ξRd\BR,

(2.1)

where ξ=|ξ|θ and, for θSd−1,

ak(θ) = ak[w](θ) :=

1 π

r

Z

−r

w(tθ)

r2t2dt, if k= 0, 2

π

r

Z

−r

w(tθ)Tk t r

r2t2 dt, otherwise.

(2.2)

In the above, (Tk)k∈N stand for the Chebyshev polynomials on R, which can be defined by Tk(t) := cos(karccos(t)) if t [−1,1] and extended to |t| > 1 in a natural way. For n= 0, the sum in (2.1) is taken to be 0. Note that formulas (2.1) and (2.2) are correctly defined for almost allξ and θ under the assumption that w∈ L(Br).

Suppose w ≈ Fv|Br. The transforms CR,nw on BR can be considered as a family of reconstruction procedures for Problem 1.2. The transforms F−1CR,nw on Rd can be considered as a family of reconstruction procedures for Problem 1.1.

In Section 3, we give stability estimates for Problem 1.1 arising from the reconstruc- tionsF−1CR,n; see Theorem 3.1 and Theorem 3.2. In Section 5, we give stability estimates

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for Problem 1.2 arising from the extrapolations CR,n; see Lemma 5.1, Theorem 5.2, and Corollary 5.4.

3 Stability estimates for Problem 1.1

We will assume that the unknown function v : Rd C satisfies (1.1) for some N, σ > 0 and ν 1 and the given data w is such that, for some δ, r >0,

kw− FvkL(Br) δ < N, (3.1) where F is the Fourier transform. Note that if (1.1) holds then, for any ξRd,

|Fv(ξ)| ≤ 1 (2π)d

Z

Rd

|v(x)|dx=Qv(0) N. (3.2) This explains the condition δ < N in assumption (3.1). Indeed, if the noise level δ is greater thanN then the given datawtells aboutv as little as the trivial functionw0 0.

To achieve optimal stability bounds, the parameters R and n in the reconstruction F−1CR,n have to be chosen carefully depending on N, δ, r, σ. For any τ [0,1], let

Lτ(δ) =Lτ(N, δ, r, σ, ν) := max (

1, 1 2

(1τ) lnNδ σrν

!τ)

. (3.3)

Here and thereafter, we assume 0< δ < N. Using (2.1), define

Cτ,δ :=CRτ(δ),nτ(δ), (3.4)

where

Rτ(δ) = Rτ(N, δ, r, σ, ν) :=rLτ(δ),

nτ(δ) = nτ(N, δ, r, σ, ν) :=

&

(2τ) ln Nδ ln 2 + τ ν1 ln(2Lτ(δ))

'

, if τ > 0,

0, otherwise.

(3.5)

and d·e denotes the ceiling of a real number. Let c(d) :=

Z

∂B1

1dx= d/2

Γ(d2 + 1). (3.6)

Our first result is a stability estimate for Problem 1.1 in the normL(Rd). In addition to (1.1), we assume also that v ∈ Wm(Rd), where the space Wm(Rd), m 0, and its

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norm are defined by

Wm(Rd) :=

u∈ L1(Rd) : (1 +|ξ|2)m2Fu∈ L(Rd) ,

||u||Wm(Rd):=

(1 +|ξ|2)m2 Fu

L(Rd).

We note that for integer m the space Wm(Rd) contains the standard Sobolev space Wm,1(Rd) of m-times smooth functions in L1 on Rd.

Theorem 3.1. Let the assumptions of (1.1) and (3.1) hold for some N, σ, r, δ > 0 and ν 1. Assume also that v ∈ Wm(Rd), for some real m > d, and that kvkWm(Rd) γ1. Then, for any α such that 0α1, the following estimate holds:

v− F−1Cτ,δ w

L(Rd) 8c(d)d N1−αrd(Lτ(δ))d+1δα +m−dc(d)γ1r−m+d(Lτ(δ))−m+d,

(3.7)

where τ = 1p

1(1α)ν−1 andLτ(δ), Cτ,δ , c(d) are defined by (3.3), (3.4), (3.6). In particular, for any β1 such that 0< m−dβ1 <1

1ν−1, we have v− F−1Cτ,δ w

L(Rd) c1 ln(3 +δ−1)−β1

, (3.8)

where τ = β1

m−d and c1 =c1(N, σ, ν, r, m, γ1, d, β1) is a positive constant.

Our second result is a stability estimate for Problem 1.1 in the norm Hs(Rd). Recall that the Sobolev space Hs(Rd),sR, and its norm can be defined by

Hs(Rd) :=

u∈ L2(Rd) : F−1(1 +|ξ|2)s2Fu∈ L2(Rd) ,

||u||Hs(Rd):=

F−1(1 +|ξ|2)s2Fu L2

(Rd).

Theorem 3.2. Let the assumptions of (1.1) and (3.1) hold for some N, σ, r, δ > 0 and ν 1. Assume also that v ∈ Hm(Rd), for some real m ≥ −d

2, and that kvkHm(Rd) γ2. Then, for any α[0,1] and any s < m, the following estimate holds:

kv− F−1Cτ,δ wkHs(Rd) 8(2π)d/2c(d)N1−α

Z rLτ(δ) 0

(1 +t2)std−1dt

!1/2

Lτ(δ)δα +γ2r−m+s(Lτ(δ))−m+s,

(3.9)

where τ := 1p

1(1α)ν−1 and Lτ(δ), Cτ,δ , c(d) are defined by (3.3), (3.4), (3.6).

In particular, for any β2 such that 0< β2

m−s <1

1ν−1, we have v− F−1Cτ,δ w

Hs

(Rd)c2 ln(3 +δ−1)−β2

, (3.10)

where τ = β2

m−s and c2 =c2(N, σ, ν, r, m, s, γ2, d, β2) is a positive constant.

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The proofs of Theorems 3.1 and 3.2 are given in Section 6. The first terms of the right-hand side in estimates (3.7) and (3.9) correspond to the error caused by the H¨older stable extrapolation of the noisy data w fromBr toBRτ(δ) and the second (logarithmic) terms correspond to the error caused by ignoring the values of Fv outside BRτ(δ); see Section 6 for more details.

Let N, σ, ν, r, m, γ1, γ2, d be fixed. Then estimates (3.8) and (3.10) used for v :=v1v2 and w:=w0 0 yield the following corollary.

Corollary 3.3. Let v1 and v2 be such that v :=v1v2 satisfies (1.1) for some N, σ > 0 and ν 1. Let τ be such that 0< τ <1

1ν−1. Then the following bounds hold.

(a) If v1v2 ∈ Wm(Rd), for some real m < d, and kv1v2kWm(Rd) γ1, then kv1v2kL(Rd)c1

ln

3 + 1

kFv1− Fv2kL(Br)

−β1

, (3.11)

where β1 =τ(md) and c1 is the constant of (3.8).

(b) If v1v2 ∈ Hm(Rd) for some real m≥ −d

2, kv1v2kHm(Rd)γ2, and s < m, then kv1v2kHs(Rd)c2

ln

3 + 1

kFv1− Fv2kL(Br)

−β2

, (3.12)

where β2 =τ(ms) and c2 is the constant of (3.10).

One can see that the estimates of Theorem 3.1, Theorem 3.2, and Corollary 3.3 are available for any β1, β2 such that:

0< β1 < β1max :=

1

1ν−1

(md), 0< β2 < β2max :=

1

1ν−1

(ms).

(3.13)

In particular, for ν = 1, we have that β1max =md and β2max =ms. In Section 4, we present instability examples showing that

if ν = 1 then (3.11) is impossible for β1 > m and (3.12) is impossible for β2 > m (when d2 ands = 0) and for β2 > m+12 (when d= 1 ands = 0);

if ν = 2 then (3.11) and (3.12) (withs = 0) are impossible forβ1, β2 > m/2.

These examples show that the logarithmic bounds (3.11) and (3.12) are rather optimal with respect to the values of the exponents β1 and β2. Consequently, in this respect, it

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is impossible to essentially improve the stability bounds of Theorems 3.1 and 3.2, even using any other reconstruction procedure (for example, based on a more advanced basis instead of Chebyshev polynomials).

Moreover, observe that, for the case ν= 1, the values of β1max and β2max are very close to the best possible: for example, we determined that β2 = m is indeed the threshold value for (3.12) when d 2 and s = 0. However, we do not know whether the claimed exponentsβ1max, β2max= 1

1ν−1

m+O(1) are also that close to optimal forν > 1.

Our instability examples for ν = 2 imply that they can not exceedm/2, but there is still a gap from m/2 down to

1 1

2

m.

Theorem 3.1, Theorem 3.2, and Corollary 3.3 illustrate similar stability behaviour in more complicated non-linear inverse problems. In fact, the relationship is closer than a mere illustration taking into account that the monochromatic reconstruction from the scattering amplitude in the Born approximation is reduced to Problem 1.1. In particular, estimates (3.10), (3.11) and (3.12) (with ν = 1) should be compared with the results on the monochromatic inverse scattering problem obtained by H¨ahner, Hohage [4], Isaev, Novikov [7, Theorem 1.2] and Hohage, Weidling [5] under the assumption that v is a compactly supported sufficiently regular function on R3. More precisely, for this case, estimate (3.11) with m >3 and β1 = m−3

3 is similar to [7, Theorem 1.2]; estimate (3.12) with s= 0, m > 32, and β2 = m

m+3 is similar to [4, Theorem 1.2]; estimates (3.10), (3.12) withm >7/2 and some appropriateβ2 (0,1) are similar to [5, Corollary 1.4]. For other known results on logarithmic and H¨older-logarithmic stability in inverse problems, see also Alessandrini [1], Bao et al. [2], Isaev [6], Isakov [9], Novikov [11], Santacesaria [13]

and references therein.

As observed above, logarithmic and H¨older-logarithmic stability was established for many different inverse problems. However, to our knowledge, even for the compactly supported case, the estimates of Theorem 3.1, Theorem 3.2 (withα <1) and Corollary 3.3 are implied by none of results given in the literature before the recent work [8]. The related results of [8] are essentially equivalent to the special case of (3.9), (3.10), (3.12) when v ∈ Hm(Rd) is compactly supported, m is a positive integer, and s= 0.

4 Examples of exponential instability for Problem 1.1

First, we recall the results from [8, Section 6]. Let A and B be open bounded domains in Rd, d 1. Then, for any fixed positive integer m and positive γ, we give examples of

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real-valued functions vn Cm(Rd) such that

supp(vn)A, kvnkCm(Rd)γ, (4.1) and the following asymptotics hold asn +∞:

kFvnkL(B) =O(e−n), kvnkL(Rd) = Ω(n−m), kvnkL2(Rd) =

Ω(n−m), for d2, Ω(n−m−12), for d= 1.

Recall that for two sequences of real numbers an and bn, we say an = Ω(bn) if an > 0 always and bn =O(an). It follows from the above that, for any β1 > mand any constant c1 >0,

kvnkL(Rd)> c1 ln

3 +kFvnk−1L(Br)

−β1

, (4.2)

when n is sufficiently large. Similarly, for any β2, whereβ2 > m if d2 and β2 > m+12 if d= 1, and for any constant c2 >0, we have that

kvnkL2(Rd) > c2 ln

3 +kFvnk−1L(Br)

−β2

, (4.3)

when n is sufficiently large.

Condition (4.1) and instability estimates (4.2) and (4.3) show an optimality (or nearly optimality) of the exponentβ1in stability estimates (3.7), (3.8), (3.11) and of the exponent β2 in stability estimates (3.9), (3.10), (3.12) with s = 0. Recall that Theorem 3.1, Theorem 3.2, and Corollary 3.3 require β1 < β1max, β2 < β2max, where β1max and β2max are defined in (3.13). In particular, for ν = 1 (which includes the compactly supported case), we have thatβ1max=md andβ2max =m (fors= 0), which are close to the infima of the exponents β1 and β2 in (4.2) and (4.3).

However, β1max(ν) and β2max(ν) decrease to 0 as ν +∞. In particular, for ν no- ticeable greater than 1, the instability behaviour exhibiting by the functions vn recalled above become much less tight with respect to β1max(ν) and β2max(ν). This motivates us to construct other explicit examples of exponential instability for Problem 1.1, which are non-compactly supported and provide considerably smaller exponents β1 and β2 in the instability estimates in comparison with vn.

For k Rd, integerm >0, and real ε >0, consider the functions vk,m,ε defined by vk,m,ε(x) :=ε|k|−me−x2/2cos(kx), xRd. (4.4)

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Note that

Fvk,m,ε(ξ) = 1

2(2π)d/2ε|k|−m

e−(ξ−k)2/2+e−(ξ+k)2/2

, ξ Rd.

Similarly to Remark 1.1, we find that the functions vk,m,ε satisfy (1.1) for ν = 2, any σ > 1

2, and N = ε|k|−mN0(d, σ). Then, for any fixed σ > 1

2, r > 0, integer m > 0, and real γ0, γ1, γ2 >0, we have that

N γ0, kvk,m,εkWm(Rd) γ1, kvk,m,εkHm(Rd) γ2; (4.5) for all sufficiently small ε > 0 and |k| > 1; and, for fixed ε > 0, the following formulas hold as |k| →+∞:

kFvk,m,εkL(Br) =O exp(−α|k|2)

, for any α (0,1

2), kvk,m,εkL(Rd) =ε|k|−m, kvk,m,εkL2(Rd) = Ω(|k|−m).

(4.6) It follows from (4.6) that, for fixed scaling parameter ε, exponents β1, β2 > m/2, and constants c1, c2 >0,

kvk,m,εkL(Rd)> c1 ln

3 +kFvk,m,εk−1L(Br)

−β1

, (4.7)

kvk,m,εkL2(Rd)> c2 ln

3 +kFvk,m,εk−1L(Br)

−β2

, (4.8)

when |k| is sufficiently large. Condition (4.5) and instability estimates (4.7), (4.8) show nearly optimality of the exponent β1 in stability estimates (3.7), (3.8), (3.11) and of the exponentβ2 in stability estimates (3.9), (3.10), (3.12) withs = 0, for the case whenν = 2.

Namely, for this case, β1max(2) =

1 1

2

(md) and β2max(2) =

11

2

m (for s= 0).

One can see thatm/2, which is the infima of the exponentsβ1 and β2 in (4.7) and (4.8), is substantially closer toβ1max(2) and β2max(2) than the infima of the exponents β1 and β2 in (4.2) and (4.3), respectively.

It is also important to note that the instability behaviour exhibiting by the functions vk,m,ε defined in (4.4) is impossible for the compactly supported case, at least for suffi- ciently large m. This is because β1max(1) and β2max(1) (for s= 0) get bigger than m/2 so (4.7) and (4.8) would contradict to Corollary 3.3.

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5 Stability estimates for Problem 1.2

Lemma 5.1. Letv ∈ L1(Rd)andλ, r, R >0be such thatrR λ/2, andQv(λ)<+∞.

Then, for any w∈ L(Br) and nN, the following estimate holds:

kFv− CR,n[w]kL(BR)2

2R r

n

kw− FvkL(Br)+ 4Qv(λ)

R λ

n

.

Lemma 5.1 is proved in Section 7. Optimising the parameter n in Lemma 5.1, we obtain the following H¨older stability estimate for Problem 1.2.

Theorem 5.2. Let v ∈ L1(Rd) and λ, r, R > 0 be such that r R λ/2, and Qv(λ) <

+∞. Suppose that kw− FvkL(Br) δ for some function w and 0< δ < Qv(λ). Then the following estimate holds:

kFv− CR,nwkL(BR) 8R

r

Q

v(λ) δ

τ(λ)

δ, (5.1)

where

n :=

ln

Qv(λ) δ

ln(2λ/r)

and τ(λ) := ln(2R/r)

ln(2λ/r). (5.2)

Remark 5.3. Problem 1.2 is a particular case of the problem of stable analytic con- tinuation; see, for example, Demanet, Townsend [3], Lavrent’ev et al. [10, Chapter 3], Tuan [14], and Vessella [15]. In particular, [3, Theorem 1.2] or [15, Theorem 1] lead to a older stability estimate similar to (5.1). In the present work, we independently establish estimate (5.1) mainly for the purpose to give a simple explicit expression for the factor in front of the H¨older term δ1−τ(λ). Besides, we derive our estimates for specific analytic functions which are the Fourier transforms of functions satisfying (1.1).

Proof of Theorem 5.2. By the assumptions, we have that

η:=

ln

Qv(λ) δ

ln(2λ/r) >0.

Note thatη is the solution of the equation 2R

r

η

δ =Qv(λ)

R λ

η

.

Using also that Rr, we get 2R

r

η+1

δ = 2R

r Qv(λ)

R λ

η

2Qv(λ)

R λ

η

.

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By definition of n and η, we find that η n < η + 1. Then, applying Lemma 5.1, we obtain that

kFv− CR,n[w]kL(BR) 2

2R r

n

δ+ 4Qv(λ)

R λ

n

2

2R r

η+1

δ+ 4Qv(λ)

R λ

η

42R

r

η+1

δ.

Then, by the definitions of τ(λ) and η, we get 2R

r

η

= exp

ηln2Rr

= exp

η τ(λ) lnr

=Q

v(λ) δ

τ(λ)

.

Combining the above formulas completes the proof.

Theorem 5.2 leads to the following stability estimate for the extrapolation Cτ,δ used in Theorem 3.2.

Corollary 5.4. Let the assumptions of (1.1) and (3.1) hold for some N, σ, r, δ > 0 and ν 1. Then, for any τ [0, ν−1], we have

kFv− Cτ,δ [w]kL(B(δ))8Lτ(δ)

N δ

ντ(2−τ)

δ = 8Lτ(δ)N1−αδα,

where Lτ(δ) and Rτ(δ) are defined in (3.3) and (3.5) and α=α(τ) := 1ντ(2τ). In addition, the exponent α is positive if and only if 0τ <1

1ν−1 ν−1.

Proof. First, we consider the caseLτ(δ) = 1. Then, (3.5) and (2.1) imply that Rτ(δ) = r and Cτ,δ [w]w. Recalling from (3.1) thatδ < N , we find that

kFv− Cτ,δ [w]kL(B(δ)) =kFv wkL(Br)δ 8Lτ(δ)

N δ

ντ(2−τ)

δ.

Next, suppose that

Lτ(δ) = 12

(1−τ) lnNδ σrν

τ

>1.

This is only possible when τ 6= 0. Let

λ:=r(2Lτ(δ))ντ1 . Then, from (1.1), we get

Qv(λ)Nexp(σλν) = δN

δ

2−τ .

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